MathPsych 2023
1University of Heidelberg, 2University of Mainz
Pros
Cons
Pros
Cons
\[\begin{align} X_{i,t}^{(w)} &= \Phi_{X,i} X_{i,t-1}^{(w)} + \beta_{YX,i} Y_{i,t}^{(w)} + \zeta_{Xi,t} \\ Y_{i,t}^{(w)} &= \Phi_{Y,i} Y_{i,t-1}^{(w)} + \Phi_{XY,i} X_{i,t-1}^{(w)} + \zeta_{Yi,t} \\ \zeta_{Xi,t} &\sim \mathcal{N}(0, \Psi_{X,i}^2), \, \zeta_{Yi,t} \sim \mathcal{N}(0, \Psi_{Y,i}^2) \end{align}\]
\[\begin{align} X_{i,t}^{(w)} &= \Phi_{X,i} X_{i,t-1}^{(w)} + \beta_{YX,i} Y_{i,t}^{(w)} + \zeta_{Xi,t} \\ Y_{i,t}^{(w)} &= \Phi_{Y,i} Y_{i,t-1}^{(w)} + \Phi_{XY,i} X_{i,t-1}^{(w)} + \zeta_{Yi,t} \\ \zeta_{Xi,t} &\sim \mathcal{N}(0, \Psi_{X,i}^2), \, \zeta_{Yi,t} \sim \mathcal{N}(0, \Psi_{Y,i}^2) \end{align}\]
\[\begin{align} X_i^{(b)} &= \gamma_1 + u_{i1} \\ Y_i^{(b)} &= \gamma_2 + u_{i2} \\ \Phi_{Xi} &= \gamma_3 + u_{i3} \\ \Phi_{Yi} &= \gamma_4 + u_{i4} \\ \Phi_{XYi} &= \gamma_5 + u_{i5} \\ \beta_{YXi} &= \gamma_6 + u_{i6} \\ \log\Psi_{Xi}^2 &= \gamma_7 + u_{i7} \\ \log\Psi_{Yi}^2 &= \gamma_8 + u_{i8} \\ \end{align}\]
\[\begin{align} \boldsymbol{u}\sim\text{MVNormal}(\boldsymbol{0}, \boldsymbol{\Omega}) \end{align}\]
real mu_X = gamma[1] + u[i,1];
real mu_Y = gamma[2] + u[i,2];
real phi_X = gamma[3] + u[i,3];
real phi_Y = gamma[4] + u[i,4];
real phi_XY = gamma[5] + u[i,5];
real beta_YX = gamma[6] + u[i,6];
real psi_X = sqrt(exp(gamma[7] + u[i,7]));
real psi_Y = sqrt(exp(gamma[8] + u[i,8]));
u[i] ~ multi_normal(rep_vector(0, 8), Omega);\[\begin{align} X_i^{(b)} &= \gamma_1 + u_{i1} \\ Y_i^{(b)} &= \gamma_2 + u_{i2} \\ \Phi_{Xi} &= \gamma_3 + u_{i3} \\ \Phi_{Yi} &= \gamma_4 + u_{i4} \\ \Phi_{XYi} &= \gamma_5 + u_{i5} \\ \beta_{YXi} &= \gamma_6 + u_{i6} \\ \log\Psi_{Xi}^2 &= \gamma_7 + u_{i7} \\ \log\Psi_{Yi}^2 &= \gamma_8 + u_{i8} \\ \end{align}\]
\[\begin{align} \boldsymbol{u}\sim\text{MVNormal}(\boldsymbol{0}, \boldsymbol{\Omega}) \end{align}\]
\(\rightarrow\) treat missing data like parameters
Model 2, simulated data convergence.
Model 2, simulated data. Errorbars: 95% CI.
Model 2, COGITO data convergence.
Model 2, COGITO data parameters. Errorbars: 95% CI.
Questions?
Github repo with Model 2 in Stan + presentation
dlsem in R:
ctsem in R:
SAS, Stata, OpenMX